Heat transfer analysis of Maxwell tri-hybridized nanofluid through Riga wedge with fuzzy volume fraction

This contribution aims to optimize nonlinear thermal flow for Darcy-Forchheimer Maxwell fuzzy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {{\text{Al}}_{{2}} {\text{O}}_{{3}} + {\text{Cu }} + {\text{TiO}}_{{2}} {\text{/EO}}} \right)$$\end{document}Al2O3+Cu+TiO2/EO tri-hybrid nanofluid flow across a Riga wedge in the context of boundary slip. Three types of nanomaterials, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{Al}}_{{2}} {\text{O}}_{{3}} ,$$\end{document}Al2O3, Cu and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{TiO}}_{2}$$\end{document}TiO2 have been mixed into the basic fluid known as engine oil. Thermal properties with the effects of porous surface and nonlinear convection have been established for the particular combination \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {{\text{Al}}_{{2}} {\text{O}}_{{3}} + {\text{Cu}} + {\text{TiO}}_{{2}} {\text{/EO}}} \right){.}$$\end{document}Al2O3+Cu+TiO2/EO. Applying a set of appropriate variables, the set of equations that evaluated the energy and flow equations was transferred to the dimensionless form. For numerical computing, the MATLAB software's bvp4c function is used. The graphical display is used to demonstrate the influence of several influential parameters. It has been observed that flow rate decay with expansion in porosity parameter and nanoparticles volumetric fractions. In contrast, it rises with wedge angle, Grashof numbers, Darcy-Forchheimer, nonlinear Grashof numbers, and Maxwell fluid parameter. Thermal profiles increase with progress in the heat source, nanoparticles volumetric fractions, viscous dissipation, and nonlinear thermal radiation. The percentage increases in drag force for ternary hybrid nanofluid are 13.2 and 8.44 when the Modified Hartmann number takes input in the range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.1 \le {\text{Mh}} \le 0.3$$\end{document}0.1≤Mh≤0.3 and wedge angle parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.1 \le m \le 0.3$$\end{document}0.1≤m≤0.3. For fuzzy analysis, dimensionless ODEs transformed into fuzzy differential equations and employed symmetrical triangular fuzzy numbers (TFNs). The TFN makes a triangular membership function (M.F.) that describes the fuzziness and comparison. This study compared nanofluids, hybrid nanofluids, and ternary nanofluids through triangular M.F. The boundary layer flow caused by a wedge surface plays a crucial role in heat exchanger systems and geothermal.


List of symbols v and u
Velocity components (m/s) is a non-Newtonian representation that displays stress relaxation behaviors.Scientists are eager to discover more about its distinctive qualities.Because of the broad array of uses that eventuate in engineering, industry, and natural processes such as cooling of electronic equipment, human transpiration, some biological fluids, DNA suspension, escalation of chemicals in plants or medicine, aerodynamic extrusion, and many others, it has gained remarkable significance and expanded motivations or consciousness between many scientists to do work in the area over the last couple of years.Well-known researchers in recent research work have made some remarkable contributions towards Maxwell nanofluid.Bilal et al. 1 examined the radiation heat flux of MHD Maxwell fluid over an upper-connected surface.Jamshed et al. 2 studied the various aspects of MHD Maxwell nanofluids.The chemical reactions on Maxwell flow via permeable surface under the effect of radiations and multiple slips have been inspected by Ali et al. 3 .Abdal et al. 4,5 reported PST and PHF properties on MHD Maxwell fluid-containing living organisms.Bilal et al. 6 analyzed the numerical simulation of the time-dependent Maxwell flow of nanofluids inspired by melting heat, magnetic fields, and Fourier and Fick laws.The liquid temperature drops when compared to melting heat and unsteadiness parameters, according to this study.Tlili et al. 7 evaluated the flow of a 3D solutal and thermal stratification on Maxwell nanofluid with a chemical process.Yahya et al. 8 studied the heat transfer analysis on Maxwell hybrid (SiO 2 + TiO 2 / Kerosene oil) nanofluid over a Riga wedge with a viscous dissipation effect.Babitha et al. 9 estimated the flow of MHD on fractional Maxwell nanofluids suspended with SWCNT and MWCNT.Several investigators have disclosed various characteristics of the Maxwell liquid 10, 11 and 12 .Darcy's law, which describes how liquid flows via a porous channel, is when Darcy-Forchheimer gets its name.This law was established based on the findings of an investigation of flowing water over the sand layer.Reynolds number variations in the porous medium, distinguished by strong inertial forces, cause movement.The significant usage of the Darcy law in petroleum technologies, grain storage, groundwater, and oil asset contexts makes it essential to study fluid mechanics.In 1856, Darcy 13 was the first researcher to claim that liquid may pass through a porous surface.Unfortunately, this idea could not be as well-known due to its limits of a slower pace and lower porosity.Forchheimer 14 changed the equation of motion by substituting the quadratic velocity requirement with the Darcian velocity to show the obvious deficiency.The high Reynolds number led Muskat 15 to coin the phrase "Forchheimer word" to describe it.To implement the Darcy-Forchheimer model to permeable media outside of the linearly enlarged zone, Pal and Mondal 16 assumed that as the electric field value increases, the concentration distribution diminishes.The flow of the MHD nanofluid via the Darcy-Forchheimer media platform as a result of the second-order boundary condition is computed by Ganesh et al. 17 .Darcy-Forchheimer law, homogeneous/heterogeneous reactions, and carbon nanotubes were used by Alshomrani et al. 18 .The Darcy-Forchheimer effect was assessed over a curving surface by Saif et al. 19 .Seth et al. 20 computed the flow of carbon nanotubes in a moving frame across a porous Darcy-Forchheimer environment.Several researchers have worked on Darcy-the Forchheimer law in the references 21,22 and 23 .
The insert of nanoparticles into a base fluid has revolutionized the field of fluid dynamics.Single-type nanoparticles are combined with the base fluid to create a nanofluid.Water, oils, ethylene glycol, and synthetic fluids are commonly used to disseminate nanoparticles.These fluids are not capable of improving heat transfer as compared to nanofluids.Metallic nanoparticles such as silver, gold, titanium, iron oxide, and aluminum are widely utilized in the base fluid.These fluids are engaged as coolants in heat transfer equipment like pharmaceutical processes, heat exchangers, engines, power plants, radiators, and electrical devices, among other things.Choi 24 first used nanoparticle dispersion in a host fluid to improve its thermal characteristics.Shafiq et al. 25 and Siddique et al. 26,27 have significantly improved base fluids' thermal properties.A hybrid nanofluid is a solution of two different types of nanoparticles in a base fluid with a higher thermal conductivity.Hybrid nanofluids simultaneously increase the chemical and physical properties of constituents.Compared to ordinary fluids and nanofluids, hybrid nanofluids significantly impact optimizing heat transfer since they have higher thermal efficiency and can be molded to meet specific needs.The effects of heat-resistive flow on MHD Sakiadis hybrid nanofluid flow through a thin needle were examined by Tlili et al. 28 .Roy et al. 29 tested the particular solution of Stoke's second theorem with the flow of a hybrid nanofluid.Shehzad et al. 30 evaluated heat conduction in a radiative hybrid nanofluid through a permeable zone.Acharya-et al. 31 explored the role of radiation on hybrid nanofluid Fe 3 O 4 /graphene flow via a folded surface.Said et al. 32 reviewed the newest advances in solar thermal collectors, emphasizing the use of nanofluids as heat transfer fluids to improve the devices' performance.Said et al. 33 examined current developments in phase change material nanotechnology with an emphasis on solar energy applications.The thermal properties of a Williamson hybrid nanofluid (MoS2 + ZnO) based on hydraulic fluid flowing across a stretched sheet were examined by Yahya et al. 34 .The numerical treatment of fuzzy hybrid nanofluid flow and heat transfer over a surface was investigated by Nadeem et al. 35,36 .
Ternary hybrid nanofluid is a novel type of fluid that leads regular fluids, nanofluid, hybrid nanofluid, acetone, and gasoline at energy exchanges.Ternary Hybrid NFs are used in heat pumps, solar energy, heat exchangers, the auto industry, air purifiers, electrical chillers, broadcasters, ships, turbines, nuclear networks, and biotechnology.Adun et al. 37 discussed ternary hybrid nanofluid's stability, heat transfer, environmental factors, and synthesis.Sundar et al. 38 addressed irreversibility generation and heat transport in an ethylene glycol-based ternary hybrid nanofluid.Ahmed et al. 39 considered the rise of heat transmission for ternary hybrid nanofluid over a square channel.Their findings proved that ternary hybrid nanofluids are better for thermophoresis and nano-cooling growth.Arif et al. 40 considered the performance of heat transmission and flow for various shaped nanoparticles based on ternary hybrid nanofluids.Gul and Saeed 41 addressed enhancing the thermal flow of the couple stress ternary hybrid TiO 2 + CoFe 2 O 4 + MgO/H 2 O nanofluid over a nonlinear stretching sheet with the Darcy-Forchheimer law.
In dynamical systems, different kinds of fuzziness or uncertainties happen related to measuring errors, material properties, environmental factors, incomplete knowledge, dimensional tolerances, comparison, engineering parameters, initial and boundary conditions, etc.This fuzziness or uncertainties will undoubtedly affect the dynamic systems, which might change the result because of the dynamic responses.In fluid dynamics, the engineering and heat transfer parameters, like nanoparticle volume fraction, exist in the governing equations.These are neither measured exactly nor their specific nominal values.So, in actual practice, these values are fuzzy or uncertain because their given information is incomplete, vague, or imprecise.In this situation, fuzzy sets theory (FST) is a useful tool for the phenomena under consideration, and it is more accurate than assuming physical difficulties.More precisely, the FDEs play a significant role in reducing the uncertainty and finding the proper way to describe the physical problem that arises in uncertain heat transfer parameters initial and boundary conditions.In 1965, Zadeh 42 gave an alternate idea of set theory, which is named FST, and this approach handles imprecise or uncertain information.The notion of fuzzy number (FN) was presented by Chang and Zadeh 43 .Further, these numbers were generalized by Dubois and Prade 44 .Different types of FNs can be categorized as triangular, trapezoidal, and Gaussian fuzzy numbers.Here, we consider TFNs for the sake of completeness.In 1987, Seikkala 45 introduced the concept of fuzzy differentiability.Later on, Kaleva 46 presented fuzzy differentiation and integration.Kandel and Byatt 47 introduced the FDEs in 1987.Several researchers have used FDEs in fluid mechanics and have found positive results.Nadeem et al. 48investigated fuzzy ohmic heating on the MHD third-grade fluid in an inclined channel.Poiseuille and Couette flows were used to study third-grade fluid in a fuzzy atmosphere scrutinized by Nadeem et al. 49 .The MHD flow of the third-grade fluid across two parallel plates in a fuzzy environment was reported by Siddique et al. 50.In their study, Nadeem et al. 49 attempted to understand how third-grade fluids behaved in Couette and Poiseuille flow when placed in a fuzzy environment.The MHD, steady, and mixed convection flow through a vertical wedge with porous material was analyzed by Kumari et al. 51 .The 2D time-dependent nanofluid flow via a vertical wedge and nonlinear mixed convection has been examined by Rajput et al. 52 .
The subsequent points demonstrate the originality of this article: i.The thermal flow rate is inflated by suspending three distinct Al 2 O 3 , Cu, and TiO 2 nanoparticles in Engine oil.ii.The transportation of nanofluids takes place through the Riga wedge, and its boundary is slipped and convected.iii.For the flow system, a Darcy--Forchheimer for the porous medium is examined.iv.The Maxwell fluid is applied.v. Nonlinear forms of thermal radiation and convection are analyzed.vi.For the fuzzy analysis, the nanoparticle volume fractions are said to be TFN controlled σ -cut.Also, the comparison of the nanofluid, hybrid, and ternary hybrid nanofluid is examined.

Mathematical formulation
Assume that steady, 2D, nonlinear mixed convection and thermal radiation on a Maxwell ternary hybrid nanofluid across a vertical wedge with Darcy-Forchheimer law.The ternary hybrid nanofluid (Al 2 O 3 + Cu + TiO 2 /EO) flows on stream coordinates (x -axis) which operate along the vertical wedge surface.In contrast, transverse coordinates y -axis are normal to the surface, as shown in Fig. 1.The calculation considers convective heat, porosity, and slip effect, where the free stream velocity is denoted as u sv = U sv x m a positive constant U sv .The boundary wall is assumed to be in motion with a velocity slip expressed as u = u v + Ru y (0).where u v > 0 rep- resents the stretching wedge, u v < 0 represents the shrinking wedge, u v = 0 represents the static wedge, and R denotes the slip coefficient.Additionally, m = ψ (2 − ψ), be the Hartree pressure gradient ψ, is the wedge angle and ̟ = ψπ determines the wedge's total angle.When the value of m is between 0 and 1, it indicates that the flow is a stagnation point flow when , whereas it is a flow past a horizontal flat surface when (̟ = 0) for m = 0 (ψ = 0).Also, T f > T ∞ , where T f and T ∞ are the surface and ambient temperatures, respectively.
The basic equations governing the steady, laminar flow of an incompressible Maxwell fluid, neglecting body forces, are where V is the velocity vector, ρ trhnf is the density of the ternary hybrid nanofluid, p is the pressure and τ is the extra stress tensor.
Constitutive equation for incompressible Maxwell fluid 1,6,8 is: is relaxation time and overhead '*' stands for the contravarient, µ trhnf is the coefficient of viscosity of ternary hybrid nanofluid and A 1 is the first Rivlin-Erickson tensor given by where superscript T denotes the transpose of the given matrix.
The primary assumption stated earlier forms the basis of the equations governing hybrid nanofluid flow, as presented in references 1,6,8 .

Boundary conditions are:
(1) ∂u ∂x In this context, it represents the velocity v in the x direction, and u denotes the velocity in the y direction.The current density of the electrodes is represented by J 0 , the heat transfer coefficient is h f , and the permanent magnet is referred to as M 0 .Table 1 summarizes the thermal properties of nanofluids, hybrid nanofluids, and ternary nanofluids.The table includes the volume percentages of Al 2 O 3 , Cu and TiO 2 nanomaterials, which are φ 1 , φ 2 and φ 3 , respectively.To convert the nanofluids, hybrid nanofluids, and ternary nanofluids to a Maxwell fluid putting φ 1 = φ 2 = φ 3 = 0 , the equation is utilized, which represents the heat capacity (ρC)p trhnf , electrical conductiv- ity σ trhnf , the liquid density ρ trhnf , specific heat Cp trhnf , dynamic viscosity µ trhnf and thermal conductivity of ternary hybrid nanofluid k trhnf .The subscripts f , nf , hnf , trhnf , s 1 ,s 2 and s 3 indicate the fluid, nanofluid, hybrid nanofluid, ternary hybrid nanofluid, and solid components of Al 2 O 3 , Cu and TiO 2 nanoparticles.Additionally, Eq. ( 14) outlines the physical properties of the engine oil Al 2 O 3 , Cu, and TiO 2 nanoparticles.

Physical interest
The skin friction Cf x and Nusslt number (Nu x ) are defined by 5,8 Then incorporate (10) to ( 15) and ( 16), resulting in the relationship shown below: Re x = u e x ν f be the local Reynolds number along the x-axis.

Fuzzification
A slight alteration in nanoparticle volume fraction can control the temperature profile of nanofluids and hybrid nanofluids in practical situations.The nanoparticle volume fraction is approached as a fuzzy number to tackle this problem using the TFN method (see Table 2).The governing ODEs are then converted into FDEs using the σ -cut .The parameter varies between 0 and 1, and it determines the degree of fuzziness of the TFN.
The approach for converting TFNs into inter val numbers σ -cut is expressed as being the key factor in the transfor- mation process, where 0 ≤ σ -cut ≤ 1.

Fuzzy Numbers
Crisp value TFN σ -cut approach

Numerical scheme
The governing flow equations of the fluid model exhibit significant nonlinearity, making it impractical to obtain exact solutions due to their high complexity.Therefore, the built-in Matlab software bvp4c numerical method is used to obtain a solution for such a problem.Also, bvp4c is a finite difference code that implements the three-stage Lobatto IIIa formula.One such method involves transforming the governing problem into a set of associated first-order equations, which is the subject of the current discussion.This is a collocation formula, and the collocation polynomial provides a C1-continuous solution that is fourth-order accurate uniformly in the interval of integration 24 .Here, the step size 0.001 is chosen to get the desired convergence criterion of 10 −6 the problem.This subsection presents the nonlinearly generated ODEs ( 11)-( 12) and the boundary constraints (13).
with boundary conditions are By utilizing the bvp4c algorithm in MATLAB, the set mentioned above of ODEs ( 20) and ( 22), along with the B.C.s (23), can be numerically solved.

Results and discussion
This research aims to use maxwell ternary hybrid nanofluids flowing over a Riga wedge for efficient cooling and heating applications in thermal engineering.The ternary hybrid nanofluid comprises solid nanoparticles such as Al 2 O 3 , Cu and TiO 2 with E.O. as a base liquid.
The solid particles dissolve in the base liquid, resulting in the formation of the tri-hybrid nanofluid.Thermal properties were found using the combination Al 2 O 3 + Cu + TiO 2 /EO of the effect of nonlinear thermal radia- tions and convection associated with the porous surface.The impact of the constraints is depicted in Figs. 2, 3,  4, 5, 6, 7, 8, 9, 10 and 11.Table 3 compares drag force and heat transmission rate with existing data from Kumari et al. 51 and Rajput et al. 52 for numerous m values to ensure the above scheme's accuracy.An exceptional match was witnessed, which validates the numerical technique and the resultant outcomes.( 20) The inspiration of the modified Hartmann number (Mh) and the dimensionless parameter (ah) on f ′ (η) is appreciated in Fig. 2. The argument is that for both fluids, velocity improves as Mh rises and falls as ah grows.The complex principles of the Mh elevate the magnetism between the wedge's boundary.Due to increased magnetism between the wedge boundary, the Riga wedge is more compact than the ordinary wedge.Also, Mh is associated with the Riga layout, which reduces and accelerates the stream's friction.The wedge parameter (m) outcomes on f ′ (η) and thermal θ(η) profiles are shown in Fig. 3, against m, and the flow rate increases the temperature drop.A higher wedge angle causes this inclination boosts fluid velocity, which forces the boundary layer's thickness to drop and the temperature to elevate.The larger value of the Maxwell fluid parameter ( ) causes the fluid motion to grow and declines the heat transfer, as publicized in Fig. 4. As it climbs, the resistance forces between the fluid particles and the surface of the wedge fall, resulting in insufficient resistance to fluid motion which rises in the flow rate and declines the heat transfer rate.
Figure 5 depicts the behaviour of velocity profiles in reaction to deviations in Grashof (β 1 ) and nonlinear Grashof numbers (β 2 ).The correlation between buoyancy and frictional forces is referred to as the Grashof number; therefore, raising the values of β 1 and β 2 reducing the viscosity behaviour of nanoparticles, resulting in a slowdown in resistive forces to fluid motion.The fluid's velocity increases in this physical phenomenon, as demonstrated in Fig. 5. Figure 6 depicts how increasing levels of the porosity parameter (Kp) and the Darcy law parameter (Fr) slow down fluid velocity.When Kp and Fr climbed, the capacity of pore space enhanced, causing  additional resistance to liquid motion and reducing the flow rate of the fluid.The velocity slip (γ ) and Biot num- ber (Bi) imprinted in Fig. 7 represent the fluid flow and temperature profiles.As demonstrated, the flow rate and thermal effects are enhanced by both velocity slip and Biot number.Velocity slip is expected to intensify fluid motion by inducing additional disruption.The reason for this is that as the velocity slip increases, the velocity of the fluid also increases.This increase in velocity results in greater applied forces, which push the expanding wedge and transfer energy to the liquid.But, it has been reported that hybrid nanofluids exhibit higher velocities than Maxwell fluids due to their improved thermophysical properties.So the Bi is used to calculate heat transfer rate; we can conclude that Bi has a direct relationship with thermal efficiency.Physically, elevating the values of the Bi improves the thermal proficiency of the fluid, causing an upsurge in the heat transmission rate.The impact of the thermal parameter (Nr) and temperature ratio parameter (θ r ) on the thermal profile is demonstrated in Fig. 8.
An increase in θ r and Nr results in an increase in the temperature field.Substantially, a larger θ r shows a superior thermal transformation among the wedge wall and the neighboring atmosphere.The presence of a radiative component increases the mobility of small particles by inducing collisions between unrelated moving particles and converting frictional energy into thermal energy.As a result, a hybrid nanofluid exhibits a higher temperature than a conventional fluid in both scenarios.The thickness of the thermal boundary layer is due to the consequential temperature variation.The presence of a radiative component increases the mobility of small  particles by inducing collisions between unrelated moving particles and converting frictional energy into thermal energy.As a result, a hybrid nanofluid exhibits a higher temperature than a conventional fluid in both scenarios.
The effect of the Eckert number (Ec) and the heat source/sink parameter (H) on the heat flux is highlighted in Fig. 9.It is acknowledged that as the Ec and H go up, the heat transfer boosts.Similarly, as equated to a Maxwell fluid, a tri-hybrid nanofluid has a faster heat transmission rate.The temperature profile is also improved due to dissipation effects in the energy equation.The main reason is that higher Ec values convert mechanical energy into thermal energy.When heat is produced, energy is transferred from the wedge to the working fluid.This increases the thermal field in the boundary layer region close to the wedge.However, as we move farther from the wedge, the temperature profile gradually reduces to zero.
In Fig. 10, we can see the impact of nanoparticle concentration on the flow and thermal fields of fluid, hybrid nanofluids, and ternary hybrid nanofluids.As the concentration of nanoparticles increases, the velocity of the fluid decreases while the thermal field improves.This occurs because the higher volume fraction of nanoparticles causes the momentum and thermal boundary layers to become denser, leading to greater resistance in the fluid.As a result, the velocity decreases.However, the presence of nanoparticles also increases the thermal conductivity of the fluid, resulting in higher fluid temperature.Furthermore, the ternary hybrid nanofluid exhibits the lowest velocity and the highest heat transfer rate compared to nanofluid and hybrid nanofluid.

Fuzzy results and discussion
The determined fuzzy temperature (θ(η, σ )) is visualized in Fig. 11 using volume fractions φ 1 , φ 2 and φ 3 as the TFN [0, 5, 10%].Four sub-plots for triangular M.F.s demonstrate the θ(η, σ ) varying values of η = 1, 2, 3, 4. The M.F. of the θ(η, σ ) for σ -cut is on the vertical axis, while the θ (η, σ ) for different values of η is on the horizontal axis.The values calculated for TFN θ(η, σ ) are not symmetrical in a triangle shape, but the fuzzy volume fraction is symmetrical and also a part of TFN.These differences may be due to the nonlinearity of the governing FDE.Furthermore, hybrid nanofluids were discovered to have a larger range than nanofluids, which results in the TFN being unable to accurately determine the hybrid nanofluid.In    4 displays the coefficient of skin friction and temperature fluctuations.Table 4 shows that skin friction is elevated via Mh, m, Nr, Ec, β 1 and β 2 while the decline via Kp and Fr.Table 4 demonstrates the numeri- cal results for the surface heat transfer rate.The increase in Mh, m, β 1 and β 2 the surface heat transfer rate for hybrid nanofluid and ternary nanofluid is prominently amplified, and the opposite results were found for rising Kp, Fr, Nr, and Ec.Observations suggest that the wall heat transfer rate is higher in a ternary nanofluid than in a hybrid nanofluid.

0. 1
≤ m ≤ 0.3 .For fuzzy analysis, dimensionless ODEs transformed into fuzzy differential equations and employed symmetrical triangular fuzzy numbers (TFNs).The TFN makes a triangular membership function (M.F.) that describes the fuzziness and comparison.This study compared nanofluids, hybrid nanofluids, and ternary nanofluids through triangular M.F.The boundary layer flow caused by a wedge surface plays a crucial role in heat exchanger systems and geothermal.
) transformed into a collection of nonlinear ODEs by employing similarity transformations (10): with boundary conditions are where Modified Hartmann number Mh = πj 0 M 0 4u 2 sv , second-grade fluid parameter = 2 1 b µ

Fig. 11 ,
nanofluids, Al 2 O 3 /EO,Cu/EO,TiO 2 /EO, hybrid nanofluids (Al 2 O 3 + Cu/EO) and ternary hybrid nanofluids (Al 2 O 3 + Cu + TiO 2 /EO) are compared using M.F. for different values of η.There are four scenarios considered in these figures.The first scenario φ 1shows as TFN and φ 2 = 0 epitomized by blue lines.The second scenario φ 2 shows as TFN and φ 1 = 0 epitomized by red lines.The third scenario shows the hybrid nanofluid with both φ 1 and φ 2 non-zero, epitomized by black lines.Ternary hybrid nanofluids φ 1 , φ 2 and φ 3 are non-zero, which represents red lines.The hybrid nanofluid performs better than both nanofluids because the temperature variance is larger in the hybrid nanofluid.In a ternary hybrid nanofluid, the thermal conductivities of Al 2 O 3 , Cu and TiO 2 are combined to achieve greater heat transfer physically.As the temperature of the fluid rises, so does its thermal conductivity.It's worth noting that the nanoparticles and temperature play a role in increasing thermal conductivity in nanofluids.The increase in temperature caused by the materials' intrinsic thermal conductivity phenomenon improves thermal conductivity.The thermal conductivity of most nanomaterials increases as the temperature rises, transferring more energy
, φ 2 and φ 3 The volume percentages of Al 2 O 3 , Cu and TiO 2 nanomaterials www.nature.com/scientificreports/Newtonian fluids are complicated fluids that a solitary relationship cannot represent.Many applications, such as ketchup, sugar solutions, apple sauce, starch suspensions, soaps, lubricants, and margarine, can display non-Newtonian fluid characteristics.A rate-type subclass called the Maxwell fluid (viscous and elastic behavior)

Table 4 .
Numerical results of ternary nanofluid and hybrid nanofluid for Cf x and Nu x .